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Math Help - Order of element

  1. #1
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    Order of element

    How do you prove that the order of a in Zn (where n is an integer greater than or equal to 1) viewed as a group of order n with respect to addition is
    n/(gcd(a,n))?
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  2. #2
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    Quote Originally Posted by PvtBillPilgrim View Post
    How do you prove that the order of a in Zn (where n is an integer greater than or equal to 1) viewed as a group of order n with respect to addition is
    n/(gcd(a,n))?
    You need to show that:

    1) a^{n/\gcd(a,n)}=0

    2)It is the smallest positive exponent which makes this statement true.
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  3. #3
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    Do you mind elaborating on this?

    By the way, I'm in a university abstract algebra course. He introduces some group theory at the end. Thanks for the help anyway.
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    Quote Originally Posted by PvtBillPilgrim View Post
    Do you mind elaborating on this?

    By the way, I'm in a university abstract algebra course. He introduces some group theory at the end. Thanks for the help anyway.
    Let G be a cyclic group (finite). And let |G|=n. It means the G has a generator that is <a>=G for some a\in G.

    Let b\in G thus, b=a^c for 1\leq c\leq n.

    We need to find the smallest m such as,
    b^m=(a^c)^m=a^{cm}=e
    By the properties of cyclic groups and the fact that a is a generator it is equivalent to saying n divides cm.
    Thus, we need the smallest m such that,
    \frac{cm}{n} is an integer.
    We can write it as, (by dividing by d=\gcd (c,n)
    \frac{m(c/d)}{(n/d)}
    But,
    \gcd \left( c/d,n/d \right)=1
    Thus,
    n/d divides m.
    The smallest such m is of course,
    n/d

    Thus, the order of any element is,
    \frac{n}{\gcd(c,n)}
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